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Chapter 6: Problem 210

Factor completely using the sums and differences of cubes pattern, if possible. $$ (x+4)^{3}-27 x^{3} $$

Short Answer

Expert verified
The factored form is \((-2x + 4)(13x^2 + 20x + 16)\).

Step by step solution

01

Identify the expression as a difference of cubes

Recognize that the expression \( (x + 4)^3 - 27x^3 \) is a difference of cubes. The general form for factoring the difference of cubes is \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2). \]
02

Assign variables a and b

Identify \( a \) and \( b \) in the expression. Here, \( a = x + 4 \) and \( b = 3x \), since \( (3x)^3 = 27x^3. \)
03

Apply the difference of cubes formula

Replace \( a \) and \( b \) in the formula. Therefore, \[(x + 4)^3 - 27x^3 = (x + 4 - 3x)((x + 4)^2 + (x + 4)(3x) + (3x)^2). \]
04

Simplify the expressions inside the parenthesis

First, simplify \( x + 4 - 3x = -2x + 4. \) Next, expand and simplify the remaining terms: \[ \begin{align*} (x+4)^2 &= x^2 + 8x + 16, \ (x + 4)(3x) &= 3x^2 + 12x, \ (3x)^2 &= 9x^2. \ \text{Now combine them:} \ x^2 + 8x + 16 + 3x^2 + 12x + 9x^2 &= 13x^2 + 20x + 16. \ \]
05

Write the final factored form

Combine all results to get the final factored form: \[ (x + 4)^3 - 27x^3 = (-2x + 4)(13x^2 + 20x + 16). \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Cubes
To solve the given problem, we start by recognizing it as a **difference of cubes**. The general formula for factoring the difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2). \]
In our exercise, we are given (x + 4)^3 - 27x^3. This expression follows the difference of cubes pattern where (a) and (b) are suitable terms that make up the cubes.
Here, we assign \( a = x + 4 \) and \( b = 3x \). This identification helps us fit the given polynomial into the classic difference of cubes framework.
Polynomial Simplification
Next in the solution, it's essential to **simplify** polynomial expressions by performing algebraic operations. For our problem, after assigning it a^3 - b^3 = (a - b)(a^2 + ab + b^2), we replace \(a \) and \(b \) with \( x + 4 \) and \( 3x \), respectively.
Following this:
\[ (x + 4)^3 - 27x^3 = (x + 4 - 3x)((x + 4)^2 + (x + 4)(3x) + (3x)^2). \]
It's crucial to simplify inside the parenthesis:- Simplify \( x + 4 - 3x \) to get \( -2x + 4 \).
- Expand \( (x + 4)^2 \ = \ x^2 + 8x + 16 \),\( (x + 4)(3x)\) becomes \( 3x^2 + 12x \), and \( (3x)^2 \ = \ 9x^2 \).Combining these results, you get \( x^2 + 8x + 16 + 3x^2 + 12x + 9x^2 \ = \ 13x^2 + 20x + 16 \).
Factoring Techniques
Several **factoring techniques** can be used to break down complex algebraic expressions. One such technique is recognizing special patterns like the difference of cubes:
  • Difference of Squares
  • Sums and Differences of Cubes.

When dealing with cubes, always watch for the \( a^3 - b^3 \) or \( a^3 + b^3 \) form. Use the difference of cubes formula \( (a - b)(a^2 + ab + b^2) \) to factorize:
  • Recognize pattern
  • Assign suitable \( a, b \)
  • Apply and Simplify.

Understanding these techniques forms the backbone of simplifying complex polynomials.
Algebraic Expressions
Finally, it's crucial to understand the different parts of **algebraic expressions**. In our example,
we start with the algebraic expression \( (x + 4)^3 - 27x^3 \),. Here, algebraic expressions combine variables, constants, and operations.
Breaking them into recognizable patterns aids in simplification:
  • Variables: x, y, z.
  • Constants: Numerical values.
  • Operations: Addition, subtraction, multiplication, exponentiation.

Recognizing patterns and practicing different types of algebraic expressions helps in mastering polynomial simplification.

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Most popular questions from this chapter

In the following exercises, factor using the 'ac' method. $$ 60 y^{2}+290 y-50 $$

In the following exercises, factor each expression using any method. $$ 24 n^{2}+20 n+4 $$

In the following exercises, factor each expression using any method. $$ (x+3)^{2}-9(x+3)-36 $$

In the following exercises, factor by grouping. $$ 16 q^{2}+20 q-28 q-35 $$

In the following exercises, factor completely using trial and error. $$ 5 y^{2}+16 y+11 $$
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